MOSFET Optimization in Deep Submicron
Technology for Charge Amplifiers
Gianluigi De Geronimo and Paul O’Connor
Abstract--The optimization of the input MOSFET for
charge amplifiers in deep submicron technologies is discussed.
After a review of the traditional approach, the impact of properly modeling the equivalent series noise and gate capacitance
of the MOSFET is presented. It is shown that the minimum
channel length and the maximum available power are not
always the best choice in terms of resolution. Also, in an optimized front-end, the low frequency noise contribution to the
Equivalent Noise Charge may depend on the time constant of
the filter. As an example, results from the commercial TSMC
0.25µm CMOS technology are reported.
1
I. INTRODUCTION
Integrated Circuits (ASICs) can
Abe a valuableSpecific
solution in front-end electronics for
II. THE ENC EQUATION
The resolution of a front-end can be measured in terms of
Equivalent Noise Charge (ENC) [4-7]. The ENC corresponds to the charge that must be delivered to the front-end
in order to achieve a signal to noise ratio equal to the unity.
In the following we will assume that the resolution of the
front-end is dominated by the noise from the input MOSFET, characterized by a gate length L, a gate width W and a
drain current density Jd=Id/W. We will also assume that the
front-end implements a time invariant filter with overall
pulse response to a charge Q equal to Q·h(t), and characterized by a maximum Q·h(t)|max.
PPLICATION
radiation sensors. Detection systems with high sensor pixellation can benefit from low power, low parasitics, high
front-end channel density, and low cost per channel. In addition the ASICs are characterized by good radiation tolerance [1-3] and can integrate large amount of additional signal processing and functions in analog, mixed-signal and
digital domains, offering further advantage in terms of
power and real estate.
In a properly designed front-end the resolution is limited
by the noise from the input transistor. Consequently a relevant phase of the design consists of optimizing the input
MOSFET with respect to sensor, interconnects and the specific application. The choice of the polarity (n- or pchannel), size (length L and width W) and operating point
(drain current density Jd and drain-to-source voltage Vds) are
determinant in achieving the best performance. The optimization process relies on equations, models and parameters
that can be strongly dependent on the technology. As deep
submicron CMOS technologies are developed and characterized for digital design, the process of optimizing the input MOSFET can become very challenging. This contribution would like to provide the low-noise front-end designers
with techniques to keep pace with the rapid evolution of
CMOS technology.
After a review of the traditional optimization process, the
impact of properly modeling the series noise and gate capacitance of the MOSFET is discussed. As an example, the
commercial TSMC 0.25µm CMOS technology is characterized and investigated for the best achievable resolution.
Input MOSFET
v n2
Q
Q·h(t)
in2
Cin
Filter
Fig. 1. Schematic of the front-end. The components relevant to the
evaluation of the ENC are shown.
In Fig. 1 a schematic of the front-end for the evaluation
of the ENC is shown. The parallel noise contribution i 2n ,
characterized by a unilateral power spectral density Sin, is
typically related to the sensor leakage current, to its bias
circuitry and to the charge amplifier feedback circuitry. In
the following we will initially assume a white density
Sin=Sp. Then, since this contribution is not related to the
input MOSFET, this term will be neglected. The series
noise contribution v 2n , characterized by a unilateral power
spectral density Svn, is dominated by noise processes in the
input MOSFET. In the following we will initially assume a
density Svn=Aƒ/ƒ+Sw. The low frequency term, known as 1/ƒ
noise, is strongly technology related. The white term is
strongly related to the transconductance gm of the MOSFET.
The input capacitance Cin includes the sensor capacitance,
the feedback capacitance and any parasitic connected to the
input node and not dependent on the size (W,L) of the input
MOSFET. The gate capacitance Cg includes any term dependent on the size (W and/or L) of the input MOSFET.
The ENC can be expressed as follows:
2
This work was supported by the US Department of Energy, Contract
No. DE-AC0298CH10886.
G. De Geronimo and P. O’Connor are with the Instrumentation Division, Brookhaven National Laboratory, Upton, NY 11973, USA (telephone: 631-344-5336, e-mail: degeronimo@bnl.gov).
Cg
ENC =
1
2π
∫ [S
∞
in
2
(
)
H ( jω) + C in + C g 2 ω2Svn H ( jω)
0
2
2
] dω
(1)
h ( t ) max
where τ is the time constant of the filter and H(s) is the
U.S. Government work not protected by U.S. copyright
Laplace transform of the pulse response h(t). Introducing
the typical spectral densities previously discussed it follows:
(Cin + Cg )
2
ENC 2 =
∞
 ∞
2
2
S w τ H( jωτ) ω2 dω + A ƒ 2π τ H( jωτ) ωdω
 0
0

2
2π ⋅ h ( t / τ) max
∫
∞
Sp
+
∫ τ H( jωτ)
2
∫



+
(2)
dω
0
2
2π ⋅ h ( t / τ) max
where the time is normalized to an arbitrary time constant τ,
and τ•H(sτ) is the Laplace transform of h(t/τ). Assuming as
time constant the pulse peaking time τp, calculated from 1%
to the peak, and transforming the integrals variable, the (2)
becomes:
S

ENC 2 = Cin + Cg 2  w a w + A ƒ 2π a ƒ  + Sp τ p a p
τ
 p

(
)
(3)
where the three ENC coefficients:
∞
aw =
∫ H( jx)
2
∞
x 2 dx
0
2π ⋅ h ( t / τ p )
, aƒ =
2
max
∫ H( jx )
2
∞
∫ H( jx)
xdx
0
2π⋅ h ( t / τ p )
2
max
, ap =
2
dx
2
(
)
CU-2
CU-3
CU-4
CU-5
CU-6
CU-7
RB-2
RB-3
RB-4
RB-5
RB-6
RB-7
CB-2
CB-3
CB-4
CB-5
CB-6
CB-7
P
4
5
6
7
8
9 10
3
4
5
6
7
8
9 10
6
8 10 12 14 16 18 20
aw
1
0.92
0.82
0.85
0.89
0.92
0.94
0.93
0.85
0.91
0.96
1.01
1.04
1.03
1.11
1.30
1.47
1.61
1.74
8 10 12 14 16 18 20
1.08
1.27
1.58
1.86
2.11
2.31
Shape
1
2
nd
th
7
0
0
1
2
3
1
2
7
0
0
nd
th
1
2
1
2
nd
0
7
th
-1
0
2
4
1
nd
2
0
th
7
-1
0
2
4
6
aƒ
0.44
0.59
0.54
0.53
0.52
0.52
0.51
0.59
0.54
0.53
0.52
0.52
0.51
0.75
0.77
0.81
0.84
0.87
0.89
ap
0.33
0.92
0.66
0.57
0.52
0.48
0.46
0.88
0.61
0.51
0.46
0.42
0.40
1.01
0.76
0.66
0.62
0.59
0.57
τw /τp
0.79
0.86
0.93
0.98
1.02
1.06
1.02
0.76
0.67
0.63
0.59
0.58
12.9
7.29
5.60
4.81
4.37
4.11
2
7.66
5.04
4.17
3.73
3.46
3.27
6.31
3.92
3.16
2.84
2.66
2.55
16.6
9.87
7.68
6.60
5.94
5.53
III. CONSTRAINTS, VARIABLES AND MODELS
max
depend only on the type of filter, and H(jx) is obtained replacing ω with x/τp in H(jω). In Table 1 the coefficients for
some commonly adopted time invariant filters are reported,
along with the ratio between the pulse width τw, calculated
from 1% to 1% of the curve, and the peaking time τp, calculated from 1% to the peak. The R-type filters have real
poles only while the C-type filters, commonly adopted in
commercial discrete shapers, have complex conjugate poles
nd
th
[8]. The order of the filters in Table 1 ranges from 2 to 7 .
It can be observed that, as the order increases, the ratio τw/τp
decreases making the high order filters more suitable for
high rate applications. Also, at equal order, the C-type filters offer a better τw/τp ratio than the R-type filters. Finally,
the advantage of zero area for bipolar filters, which in absence of effective baseline stabilization at high rate can be
relevant, is compensated by worse values in the series coefficients aw and aƒ, especially for high orders.
Without loss of generality we will simplify our analysis
by assuming aw=1, aƒ=0.5, and ap=0.5, which are close to the
typical values for a good time invariant unipolar shaper.
In the rest of the analysis the parallel noise contribution
Sp will be neglected, and the general expression for the
ENC in (3) will be simplified as follows:
S

ENC 2 = C in + C g 2  w + πA ƒ  .
 τ p

W
Filter
Triang.
RU-2
RU-3
RU-4
RU-5
RU-6
RU-7
(4)
0
2π ⋅ h ( t / τ p )
TABLE I
COMMONLY ADOPTED TIME INVARIANT FILTERS AND CORRESPONDING
ENC COEFFICIENTS. THE RATIO τ /τ IS ALSO REPORTED.
(5)
The input MOSFET must be optimized for the minimum
ENC. The ENC depends on some parameters not related to
the input MOSFET, specifically the input capacitance Cin,
the peaking time τp, and the maximum power Pd_max available
to the input MOSFET. Typically the input capacitance Cin is
set by the sensor and interconnects, the maximum power
Pd_max depends on system level constraints, and the peaking
time τp is set by the rate or ballistic deficit. In presence of
non-negligible parallel noise contribution a further constraint on the peaking time may occur.
The typical optimization process consequently assumes
Cin, τp, and Pd_max as constraints. One further constraint derives from the use of a single supply Vdd:
Id =
Pd
Vdd
⇒
Jd =
Id
W
(6)
where Jd is the MOSFET drain current density. The (6) establishes a constraint on the Jd×W product. It follows that,
once Cin, τp, and Pd are defined, the ENC depends only on
two variables: W (or Jd) and L. The optimization process
will consequently return the two optimum values Wopt (or
Jd_opt) and L_opt that give the minimum ENC (ENCopt).
In order to proceed with the optimization of the input
MOSFET, the parameters Cg, Sw and Aƒ must be expressed
as functions of W (or Jd) and L. In the following sections
the modeling of these parameters will be discussed, starting
from the solution that the CMOS designers frequently
adopted in the past, before the advent of the deep submi-
U.S. Government work not protected by U.S. copyright
120
G
W
100
ƒ
In the past the front-end designers frequently adopted the
following models [9-13]:
C g = C ox WL, S w =
Kƒ
2 4kT
, Aƒ =
3 g m (J d , L)
C ox WL
(7)
where Cox is the oxide capacitance, k is the Boltzmann constant, T is the absolute temperature, gm is the MOSFET
transconductance dependent on Jd and L, and Kƒ is the 1/ƒ
noise coefficient. We will refer to this model as C-model.
Introducing (7) in (5) it follows:

Kƒ 
4 kT
2 1 2
+π
ENC 2 = (C in + C ox WL ) 

(
)
τ
3
g
J
,
L
⋅
W
C
mw d
ox WL 

 p
(8)
Tranconductance density gmw [mS/mm]
where gmw(Jd,L) is the transconductance per unit of W.
The function gmw(Jd,L) can be easily extracted for specific
values of L from a Spice simulation using the model and
parameters available for the technology. The coefficient Kƒ
can be extracted from a measurement of the input noise
spectral density, it is assumed to be independent of L and it
may differ for n-channels (Kƒn) and p-channels (Kƒp).
2
For the TSMC 0.25µm technology Cox≈6.1fF/µm ,
gmw(Jd,L) can be obtained from BSIM3v3.1 simulations (see
Fig. 2) and, from measurements on samples with minimum
-24
-24
L at 1kHz, it follows Kƒn≈6×10 J and Kƒp≈0.3×10 J.
In Fig. 3 the optimum ENC for Cin=1pF and τp=1µs, calculated for n- and p-channels with different L as function of
Pd is shown. For minimum L the white and 1/ƒ components
and the optimum W are also reported. It can be observed
that in this case the p-channel offers a better resolution than
the n-channel (lower Kƒ) and this is generally true. It also
indicates that the choice Lopt=Lmin and Pd_opt=Pd_max offers better results in an amount that depends on the relative ratio
between the white and 1/ƒ contributions.
10
3
10
2
10
1
10
0
10
-1
10
-2
Nch
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
**
*
**
Pch
-4
10
-3
10
-2
10
-1
10
0
10
1
white
1/ƒ
white
1/ƒ
1.0
0.8
Nch
60
0.6
40
0.4
Pch
20
0
-5
10
0.2
TSMC 0.25µm - Cin=1pF, τp=1µs
10
-4
10
0.0
-3
10
-2
Power Pd [W]
Fig. 3. Simulated ENCopt vs Pd for n- and p-channels with different
channel lengths in TSMC 0.25µm adopting the C-model (7).
V. ENHANCED (E) MODEL FOR S
W
The model for Sw in (7) provides a crude white noise estimation valid for long channel MOSFETs operating in
strong inversion. In most cases the input MOSFET of a
front-end operates in moderate to weak inversion, and a
better estimate is needed. In addition, in deep submicron
MOSFETs an excess noise factor above unity has been frequently reported.
An equation that better estimates the white noise in all
regions of operation can be found in the literature [14-16]:
S 2vw = α w 4kT
µ eff Q I 1
L2
(9)
g 2m
where µeff is the mobility, QI is the total inversion charge,
and αw is the excess noise factor. This equation can be approximated with [16]:
S2vw = α w nγ(J d , L)
2kT
g mw (J d , L)W
(10)
where n≈(gmb+gm)/gm is the subthreshold slope coefficient
ranging around 1.2-1.3 and γ(Jd,L) is a coefficient ranging
between 1/2 and 2/3. The same authors of [16] proposed the
following interpolation function:
Jd L
1
1 2

+ u (J d , L ), u (J d , L) =
1 + u (J d , L )  2 3
µ eff C ox 2nVt2

(11)
where u(Jd,L) is known as inversion coefficient and
Vt=kT/q. From the same authors is a simpler approximation
for γ(Jd,L) [12]:
* moderate inversion (Vgs~VT)
γ ( J d , L) =
-3
10
80
γ ( J d , L) =
TSMC 0.25µm - BSIM3v3.1
10
ENCopt [r.m.s. electrons]
IV. CLASSICAL (C) MODEL FOR C , S AND A
1.2
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
Wopt [mm]
cron CMOS technologies.
10
2
Drain current density Jd [mA/mm]
Fig. 2. Simulated gmw vs Jd of n- and p-channels with different L in
TSMC 0.25µm. Center of moderate inversion (*) is also indicated.
2
 g mw (J d , L )

3 + 
nVT 
Jd


.
(12)
A slower interpolation function is reported in [15], along
with the companion model for the gate capacitance. A comparison to Cg simulations and to noise measurements [2]
suggests that (11), here adopted, might be more accurate. In
U.S. Government work not protected by U.S. copyright
Fig. 4 the gamma coefficient γ(Jd,L) versus the inversion
coefficient u(Jd,L) is shown. The regions of operation for
the MOSFET are also indicated. In most cases the optimum
input MOSFET operates in moderate inversion.
0.65
Gamma coefficient γ
strong
(VGS-VT > ~10Vt)
0.60
moderate
VI. ENHANCED (E) MODEL FOR CG
The gate capacitance Cg in (5) includes any term dependent on the size (W,L) of the input MOSFET. A better estimate of Cg should consider the gate-to-source and gate-todrain overlap components [15]. These extrinsic terms are
proportional to the width W of the MOSFET and, in deep
submicron technologies, are typically not negligible. In
addition, the intrinsic component of Cg in saturation is a
fraction ≈2/3 of CoxWL [15]. An equation that improves the
estimate of Cg, here referred as Ci-model, is:
0.55
C g = 2C ov W +
weak
0.50
-3
10
10
-2
10
-1
10
0
10
1
10
2
10
2
2


ENC 2 =  C in + 2C ov W + C ox WL  ×
3


Fig. 4. Gamma coefficient γ vs inversion coefficient u. The regions of
operation of the MOSFET are also indicated.
 1 α w nγ (J d , L ) 4kT
Kƒ 
ENC 2 = (C in + C ox WL )2 
+π

C ox WL 
 τ p g mw (J d , L ) ⋅ W
(13)
For the TSMC 0.25µm technology nnch≈1.2, npch≈1.3 and,
from measurements, αw≈1 whenever u<10 (weak and moderate inversion) and Vds-Vds_sat is small (Vds_sat is the drainsource saturation voltage). Good agreement with these results can be found in the literature [2,18-20].
Nch - L=0.24µm
Pch - L=0.24µm
1.0
C
0.8
60
0.6
u~1
40
0.4
20
Wopt [mm]
ENCopt [r.m.s. electrons]
120
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
E(Sw),Ci(Cg)
100
1.2
80
0.2
u~1
0
-5
10
For the TSMC 0.25µm technology typical values for Cov
are in the range 300-600 fF/mm, where the lower values are
reported by TSMC and the higher values are reported by the
MOSIS Service [21].
TSMC 0.25µm - Cin=1pF, τp=1µs
-4
-3
10
10
0.0
-2
10
Power Pd [W]
Fig. 5. Simulated ENCopt vs Pd for n- and p-channels with different
channel lengths in TSMC 0.25µm adopting the E-model for Sw. The center
of moderate inversion (u=1) is also indicated.
In Fig. 5 the optimum ENC from (13) for Cin=1pF and
τp=1µs, calculated for n- and p-channels with minimum L as
function of Pd, is shown, compared to C-model results. The
optimum W is also reported. It can be observed that the new
model provides a better estimate especially at low power,
where the white noise component dominates. A relatively
small difference in terms of optimization (Wopt) can also be
observed. The Lopt=Lmin and Pd_opt=Pd_max still appears the best
choice.
ENCopt [r.m.s. electrons]
120
(15)
 1 α w nγ (J d , L )4 kT
Kƒ 
×
+π

C ox WL 
 τ p g mw (J d , L ) ⋅ W
Introducing the new model (10)-(11) in (5) it follows:
E(Sw)
(14)
where Cov is the overlap capacitance density, equal for drain
and source. Introducing (14) in (13) it follows:
3
Inversion coefficient u
100
2
C ox WL
3
Nch
80
60
Pch
40
20
TSMC 0.25µm - Cin=1pF, τp=1µs
0
-5
10
10
-4
10
-3
10
-2
Power Pd [W]
Fig. 6. Simulated ENCopt vs Pd for n- and p-channels with different
channel lengths in TSMC 0.25µm adopting the E-model for Sw and the Cimodel for Cg.
In Fig. 6 the optimum ENC from (15) for Cin=1pF and
τp=1µs, calculated for n- and p-channels with minimum L as
function of Pd, is shown. When compared to Figs. 3 and 5 it
can be observed that the choice Lopt=Lmin does not apply
anymore. Both n-channel and p-channel can offer better
resolution for L higher than Lmin.
This conclusion is valid whenever 1/ƒ noise dominates,
and can be understood by rewriting (15) for this case:
2
Kƒ
2


ENC 2 ≈  C in + 2C ov W + C ox WL  π
3
C


ox WL
(16)
By calculating the differential of (15) with respect to W,
U.S. Government work not protected by U.S. copyright
Wopt =
2C ov
C in
2
+ C ox L
3
.
Total Gate Capacitance Density Cgw [pF/mm]
the optimum width for the minimum ENC can be calculated
for each L:
(17)
Capacitive matching applies (Cg=Cin) and, by superposition, the ENCopt can be written:
2
ENC opt
≈

3C ov  .
8

C in πK ƒ 1 +
3
C
ox L 

(18)
The plot in Fig. 7, derived from (18) for n-channel
MOSFETs, illustrates the impact of L on the optimum ENC
for the case of dominant 1/ƒ noise. Intuitively, as L increases the 1/ƒ noise contribution decreases more rapidly
than the increase in gate capacitance.
3.6
TSMC 0.25µm - BSIM3v3.1 - MOSIS
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
3.2
2.8
Nch
Pch
2.4
Ci
2.0
Nch - L=0.24µm
E
1.6
1.2
-4
10
10
-3
10
-2
10
-1
10
C
0
10
1
10
2
Drain Current Density Jd [mA/mm]
Fig. 8. Simulated (BSIM3v3.1) Cgw vs Jd of n- and p-channels with different L in TSMC 0.25µm. The C- and Ci-models for the minimum L nchannel are also reported.
70
Introducing the (19) in (13) it follows:
 1 α w nγ (J d , L )4kT
Kƒ 
ENC 2 = [C in + C ow (J d , L ) ⋅ W ]2 
+π

(
)
τ
g
J
,
L
⋅
W
C
ox WL 

 p mw d
50
assumung 1/ƒ dominant
40
1
10
Channel length L [µm]
Fig. 7. Simulated ENCopt vs L for n-channel MOSFET and Cin=1pF assuming 1/ƒ dominant.
The Ci-model (14) for Cg doesn’t take into account the
dependence of the intrinsic gate capacitance on the drain
current density Jd. This dependence in deep submicron
technologies can be not negligible, especially in the transition from weak, thorough moderate, to strong inversion. An
enhanced (E) model for Cg can be written as:
C g = C gw (J d , L ) ⋅ W
(19)
where Cgw(Jd,L) is the gate capacitance density and includes
the bias dependence, intrinsic and extrinsic components.
As for gmw(Jd,L) the function Cgw(Jd,L) can be extracted for
specific values of L from a Spice simulation using the
model and parameters available for the technology. For the
TSMC 0.25µm Cgw(Jd,L) can be obtained from BSIM3v3.1
simulations as shown in Fig. 8. The results for C- and Cimodels for the n-channel at minimum L are reported for
comparison. It is worth noting that the cutoff frequency of
the MOSFET, given by gm/(2πCg), remains an increasing
function of Jd and, in the regions of interest, typically exceeds the tens of MHz.
(20)
In Fig. 9 the optimum ENC from (20) for Cin=1pF and
τp=1µs, calculated for n- and p-channels with different L as
function of Pd, is shown. For two cases the 1/ƒ component
and the optimum W are also reported. It can be observed
again that the choice Lopt=Lmin does not apply. In addition,
for n-channel MOSFETs, the choice Pd_opt=Pd_max does not
apply, since they can offer better resolution at Pd lower than
the maximum available.
120
1.2
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
E(Sw,Cg)
100
80
60
1.0
1/ƒ
1/ƒ
0.8
Pch
0.6
Nch - L=0.48µm
40
0.4
20
0
-5
10
Wopt [mm]
60
ENCopt [r.m.s. electrons]
ENCopt [r.m.s. electrons]
TSMC 0.25µm - Nch - Cin=1pF
0.2
TSMC 0.25µm - Cin=1pF, τp=1µs
10
-4
0.0
-3
10
10
-2
Power Pd [W]
Fig. 9. Simulated ENCopt vs Pd for n- and p-channels with different L in
TSMC 0.25µm adopting the E-model for Sw and Cg.
This conclusion is valid whenever 1/ƒ noise dominates,
and can be understood by rewriting (20) for this case:
[
ENC 2 ≈ C in + C gw (J d , L )W
ƒ
.
] 2 π C KWL
(21)
ox
By taking into account (6) and calculating the differential
of (21) with respect to W, the optimum width for the minimum ENC can be calculated for each L:
U.S. Government work not protected by U.S. copyright
.
(22)
10
Capacitive matching does not apply except for the regions where ∂Cgw/∂Jd=0 and, by superposition, the ENCopt
turns out to be:

η(J d , L )C gw (J d , L )


1
2
 C in πK ƒ
ENC opt ≈ 1 +
C ox L
η(J d , L ) 
.



(
)
∂
C
J
,
L
J
gw d
η(J , L ) = 1 − 2
d
 d
∂J d
C gw (J d , L )

(23)
The plot in Fig. 10, derived from (23) for n-channel with
minimum L and Cin=1pF, illustrates the impact of Pd on the
optimum ENC for cases of dominant 1/ƒ noise. The ratio
Cg_opt/Cin is also reported. Intuitively, as Pd increases the gate
capacitance increases while the 1/ƒ noise contribution does
not change.
80
1.6
1.4
Cg_opt / Cin
ENCopt [r.m.s. electrons]
Nch - L=0.24µm, Cin=1pF
assumung 1/ƒ dominant
60
est for our applications. Some authors have reported an
increase moving towards very strong inversion [28-30].
1.2
Noise spectral density Svn [V/√Hz]
C in

∂C gw (J d , L ) 
Jd
C gw (J d , L )1 − 2

(
)
C
J
,
L
∂J d
gw d


-6
10
-7
10
-8
10
-9
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
ratio ≈2.4
slope 1/ƒ
TSMC 0.25µm
W=7.2mm, Jd=140µA/mm, Vds=400mV
10
-10
10
1
10
2
10
3
10
4
5
10
10
6
10
7
Frequency ƒ [Hz]
Fig. 11. Measured equivalent input noise spectra for n- and p-channels
with different L in TSMC 0.25µm.
10
-6
W=7.2mm, Jd=20µA/mm to 1mA/mm, Vds=200mV to 400mV
Noise spectral density Svn [V/√Hz]
Wopt =
10
-7
10
-8
10
-9
Nch - L=0.24µm
Pch - L=0.24µm
TSMC 0.25µm
-10
40
-5
10
10
-4
10
-3
1.0
10
-2
10
10
2
3
10
10
VII. ENHANCED (E) MODEL FOR LOW-FREQUENCY NOISE
In Fig. 11 the typical equivalent input noise spectral densities measured on n- and p-channels in TSMC 0.25µm with
different L are shown, and are compared to the 1/ƒ slope.
Two relevant details should be observed. The first concerns
the slope, which differs from 1/ƒ, being in this case higher
for p-channels and lower for n-channels. The second concerns the ratio between spectra, which is higher (≈2.4 for
L=0.24µm vs L=0.48µm) compared to the square root of the
ratio between L (≈1.4 for L=0.24µm vs L=0.48µm). A similar trend for short channels in deep submicron technologies
was reported by other authors [2,22-25].
In Fig. 12 the typical spectra for n- and p-channels with
L=0.24µm at different drain current densities Jd are shown.
The dependence of the low-frequency component on the
bias point appears negligible. This result seems in agreement with others in the literature for MOSFETs operating
from weak inversion up to the border between moderate
and strong inversion [3,26-29], which is the region of inter-
5
10
10
6
10
7
Frequency ƒ [Hz]
Power Pd [W]
Fig. 10. Simulated ENCopt vs Pd for minimum L n-channel and Cin=1pF
assuming 1/ƒ dominant.
4
Fig. 12. Measured equivalent input noise spectra for n- and p-channels
with minimum L and different Jd in TSMC 0.25µm.
The results reported in Figs. 11 and 12 indicate that the
low-frequency component of the noise power spectral density could be better approximated by using the following
Enhanced (E) equation:
Sƒ =
Aƒ
ƒ
αƒ
=
K ƒ (L)
C ox WLƒ
(24)
αƒ
where Kƒ(L) now depends on L, and the slope depends on
the coefficient αƒ.
The non-1/ƒ slope requires a review of the low-frequency
component ENCLF of the ENC. The second term of (2) now
becomes:
∞
(Cin + Cg )2 A ƒ (2π)α ∫ τ H( jωτ) 2 ω2−α dω
ƒ
ƒ
ENC 2LF =
0
2
,
(25)
2π ⋅ h (t / τ) max
and (3) can be rewritten:

S
(2π)α ƒ
ENC 2 = Cin + C g 2  w a w + A ƒ 1−α a ƒ α ƒ  + S p τ p a p
ƒ

 τp
τp


(
)
U.S. Government work not protected by U.S. copyright
( )
(26)
Fig. 13 for the case Cin=1pF, Pd=200µW, τp_opt =1µs.
where the ENC coefficient aƒ(αƒ) is given by:
( )
a ƒ αƒ =
∫ H( jx )
2
x
2−α ƒ
120
dx
0
2
2π⋅ h ( t / τ p )
E
.
TSMC 0.25µm - Pd=200µW, Cin=1pF, τp, opt=1µs
100
(27)
ENC [r.m.s. electrons]
∞
max
Concerning the two other coefficients in (4) it is worth
noting that aw=aƒ(0) and ap=aƒ(2). In Table 2 the ratio
ρƒ=aƒ(αƒ)/ aƒ(1) is calculated for the filters of Table 1.
TABLE II
Nch - L=0.48µm
Pch - L=0.24µm
white
white
1/ƒ
1/ƒ
80
60
40
20
RATIO ρ =aƒ(αƒ)/aƒ(1) VS αƒ FOR THE FILTERS REPORTED IN TABLE 1.
ƒ
Relative coeff. ρƒ
Filter
0
-7
10
1.1
10
7
0.8
α
0.9
1.0
1.1
-5
10
Fig. 13. Simulated ENC vs τp for a design optimized for Cin=1pF,
Pd=200µW, and τp_opt.=1µs.
th
0.9
-6
Peaking time τp [s]
nd
2
1.0
1.2
VIII. ACHIEVABLE RESOLUTION IN TSMC0.25µM
1.1
nd
2
1.0
7
0.9
By applying (28) to the TSMC 0.25µm CMOS it is possible to estimate the ENC achievable with this technology.
The results below will give a general idea of what to expect.
An exhaustive analysis is beyond the scope of this work.
th
α
0.8
0.9
1.0
1.1
1.2
1.1
2
1.0
nd
160
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
th
7
0.9
140
α
0.8
0.9
1.0
1.1
1.2
1.1
nd
2
1.0
7
0.9
th
α
0.8
0.9
1.0
1.1
1.2
The final expression for the ENC related to series noise,
adopting the E-model for Sw, Cg and Sƒ, becomes:
ENCopt [r.m.s. electrons]
RU-2
RU-3
RU-4
RU-5
RU-6
RU-7
CU-2
CU-3
CU-4
CU-5
CU-6
CU-7
RB-2
RB-3
RB-4
RB-5
RB-6
RB-7
CB-2
CB-3
CB-4
CB-5
CB-6
CB-7
E
120
C
100
Nch
80
60
40
Pch
20
TSMC 0.25µm - Cin=1pF, τp=1µs
ENC 2 = [C in + C ow (J d , L ) ⋅ W ]2 ×
( )
(28)
For the TSMC 0.25µm technology typical values for Kƒ
and αƒ are reported in Table 3. As in previous cases we will
simplify our analysis by assuming ρƒ=1.05 for n-channels
and ρƒ=0.95 for p-channels, values close to the typical for a
good time invariant unipolar shaper.
TABLE III
TYPICAL VALUES OF K AND αƒ FOR THE TSMC 0.25µM TEHCNOLOGY.
ƒ
Nch-0.24µm Nch-0.36µm Nch-0.48µm Pch-0.24µm
Pch-0.36µm
Kƒ
2.71×10-24
1.40×10-24
0.97×10-24
0.60×10-24
0.50×10-24
αƒ
0.85
0.85
0.85
1.08
1.08
As consequence of the non-1/ƒ slope, a front-end optimized for a peaking time τp_opt will show in the ENC a lowfrequency noise component dependent on τp, as shown in
-4
-3
10
10
-2
10
Power Pd [W]
(a)
1.2
1.0
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
E
0.8
Wopt [mm]

 1 α nγ (J , L )4kT
K ƒ (2π )α ƒ
w
d
aƒ αƒ 
×
+
1
−
α
ƒ
C ox WL τ p

 τ p g mw (J d , L ) ⋅ W


0
-5
10
C
0.6
0.4
Nch
0.2
Pch
0.0
-5
10
10
TSMC 0.25µm - Cin=1pF, τp=1µs
-4
10
Power Pd [W]
-3
10
-2
(b)
Fig. 14. Simulated (a) ENCopt vs Pd and (b) corresponding Wopt vs Pd for
n- and p-channels with different L in TSMC 0.25µm adopting the E-model.
Results from the C-model at minimum L are also shown.
U.S. Government work not protected by U.S. copyright
In Figs. 14(a) and 14(b) the ENCopt and Wopt for Cin=1pF
and τp=1µs, calculated using the E-model for n- and pchannels with different L as functions of Pd are shown. For
comparison the two minimum L cases with the C-model are
reported. The difference between E- and C-model in the
estimate of the ENCopt for the n-channels is relevant, while
for the p-channel may appear small. On the other hand the
difference in terms of optimization (Wopt) is in both cases
relevant.
1000
E
TSMC 0.25µm - Pd_max=1mW, Cin=1pF
inputs, 16 with L=0.24µm and 16 with L=0.36µm. The gate
capacitance was, in both cases, on the order of 1.4pF. We
observed a ≈40% difference in ENC slope, a result that appears in line with the discussion here presented. In Fig.
16(b) the ENC vs τp from a prototype for a Coplanar Grid
Sensor [32] is shown. The ASIC implements an n-MOS
input with L=0.36µm. The gate and input capacitances were
on the order of 12pF and 60pF respectively. A low-0.15
frequency noise component proportional to ≈τp , in
agreement with αƒ≈0.85, can be observed.
600
-3
Nch
-4
10
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
10
-8
10
Measured ENC [r.m.s. electrons]
100
Pd_opt [W]
ENCopt [r.m.s. electrons]
10
Pch
-7
10
10
-6
10
-5
Peaking time τp [s]
300
-
26e /pF
200
100
TPC Front-End - TSMC 0.25µm
Cg~1.4pF, Pd~230µW, τp~900ns
0
2
4
1000
-3
8
10
12
Nch
10
-4
Nch
10
Pch
10
E
-5
TSMC 0.25µm - Pd_max=1mW, τp=1µs
-13
10
10
-12
(a)
10k
-11
10
Input capacitance Cin [F]
(b)
Fig. 15. Simulated ENCopt and Pd_opt vs (a) τp and (b) Cin for n- and pchannels with different L in TSMC 0.24µm adopting the E-model.
In Figs. 15(a) and 15(b) the ENCopt and Pd_opt for Cin=1pF
vs τp and for τp=1µs vs Cin, calculated using the E-model for
n- and p-channels with different L, are shown, assuming a
power budget limit of Pd_max=1mW. In the case of Fig. 15(a)
the reduction in Pd_opt can be observed for higher values of
τp. In the case of Fig. 15(b) the reduction in Pd_opt can be
observed for smaller values of Cin. In both cases this is consequence of the dependence of the gate capacitance on the
drain current density, which pushes the input MOSFET
towards the weak inversion.
In Figs. 16(a) and 16(b) we report some experimental results from two front-end ASICs recently developed in
TSMC 0.25µm CMOS. In Fig. 16(a) The ENC vs Cin from a
prototype for a Time Projection Chamber [31] is shown.
The ASIC implements 32 front-end channels with n-MOS
Measured ENC [r.m.s. electrons]
100
10
Nch - L=0.24µm
Nch - L=0.36µm
Nch - L=0.48µm
Pch - L=0.24µm
Pch - L=0.36µm
Pd_opt [W]
ENCopt [r.m.s. electrons]
6
Input capacitance Cin [pF]
Pch
1
-14
10
-
37e /pF
400
0
(a)
Nch. - L=0.24µm
Nch. - L=0.36µm
500
CPG Front-End - TSMC 0.25µm
L=0.36µm, Cg~12pF, Cin~60pF, Pd~3mW
Measurement
Theoretical fitting
1k
white series
low-frequency series τp
-0.15
parallel
100
0.1
1
10
100
Peaking time τp [µs]
(b)
Fig. 16. ENC measurements on front-end ASICs for (a) a Time Projection Chamber and (b) a Co-Planar Grid Sensor.
IX. CONCLUSIONS
The optimization of the input MOSFET for charge amplifiers in deep submicron technology requires a proper modeling of the series noise and gate capacitance, and a review
of the ENC equation. The enhanced modeling and associated ENC equation presented here allow a better ENC estimate and a more accurate optimization. The analysis shows
that the traditional choice of selecting the minimum channel
length and the maximum of the available power do not always offer the best resolution. Also, for a defined front-end,
the low-frequency noise contribution to the ENC may de-
U.S. Government work not protected by U.S. copyright
pend on the time constant of the filter. The results here reported are based on the commercial TSMC 0.25µm CMOS,
but can be easily extended to other deep submicron technologies.
X. ACKNOWLEDGMENT
The authors are grateful to Giovanni Anelli (CERN),
Veljko Radeka (BNL), and Angelo Dragone (Bari Polytechnic) for helpful discussions.
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